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How to Determine the Limits of Functions Video

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  • 0:06 Properties of Limits
  • 1:00 Breaking Down the Function
  • 3:16 Continuous Functions
  • 4:48 Lesson Summary
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Lesson Transcript
Instructor: Zach Pino
You know the definition of a limit. You know the properties of limits. You can connect limits and continuity. Now use this knowledge to calculate the limits of complex functions in this lesson.

Properties of Limits

The addition property for limits
Limits Addition Property

So let's talk for a minute about calculating limits. Let's look at the function f(x) = (x - 3)sin(x) + 10. Let's find the limit of this function as x approaches 3.7. To do this, let's first recall properties of limits. We have an addition property that might be useful here. This says that the limit as x approaches C of some sum of two functions is equal to the limit of each one of those functions taken separately and added together. This is our 'divide and conquer' property. We have another 'divide and conquer' property when looking at products. This says that if you want to take the limit of two functions that are multiplied together, you can take the limit of each one of those functions separately and multiply the answer together. Again, this is 'divide and conquer.'

Breaking Down the Function

If we go back to our function, f(x) = (x - 3)sin(x) + 10, and we want to find the limit as x goes to 3.7 of this function, we're going to look at these pieces individually. We're going to use the product rule to separate x - 3 from sin(x), we're going to use the addition rule to look at that + 10, and we're going to use the subtraction rule, which is just like the addition rule, to separate the x - 3 into x and 3. Now I'm pretty confident that if I graph out x and 3, I can show that as x goes to 3.7, x will also go to 3.7. As x goes to 3.7, 3 will go to 3. I'm also pretty confident that as x goes to 3.7, 10 will stay at 10. So I can plug in all these numbers except for this sin(x). So what do we do about that?

Using the product, addition, and subtraction rules to break down the function
Breaking Down Function

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